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In the Hall–Héroult process of extracting aluminium from aluminium oxide, the electrolyte is a liquid mixture of cryolite $(\ce{NaAlF6})$ and aluminium oxide $(\ce{Al2O3}).$ The cathodes and the anodes are made of graphite.

During the electrolysis process, aluminium is deposited at the cathode and oxygen is liberated at the anode. Some of this oxygen reacts with the carbon in the graphite to form carbon-dioxide, thus slowly burning away the anodes. Thus, the anodes have to replaced periodically.

But if this is the case, why do they use graphite anodes at all? Why don't they use anodes of some other, less reactive material? I'm pretty sure there are many out there that would fit the bill.

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According to this source, graphite is used as anode material because carbon is easier to oxidize than $\ce{O^2-}$. The anodic oxidation of carbon

$$\ce{C + 2O^2- \rightarrow CO2 + 4e-}$$

proceeds more efficiently than the oxidation of $\ce{O^2-}$ at an inert anode

$$\ce{2O^2- \rightarrow O2 + 4e-}$$

$\ce{C}$ is a much better reducing agent than $\ce{O^2-}$ whose oxidation state is not changed in the reaction with graphite. As a consequence, anodic current density and the deposition rate of aluminium at the cathode will increase.

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    $\begingroup$ And also given that under most conditions carbon dioxide is a gas it leaves no trace in the obtained material ($\ce{CO2 ^}$). $\endgroup$ – Martin - マーチン May 16 '14 at 16:37
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The basic reason for using graphite as an anode is in Hall–Héroult process in the electrolytic reduction of alumina to aluminum metal is because graphite being an allotrope of carbon and an inert electrode reacts with oxygen to give out carbon dioxide which thus prevents the liberation of oxygen as a final product at the anode.

Had there been any other metal oxygen would have been liberated which would have readily reacted with aluminum (as aluminum has a high affinity for oxygen in comparison to other metals, for example, iron—even if iron reacted with oxygen the aluminum will easily displace it) to form aluminum oxide and the whole process will prove to be useless.

Graphite is also a good conductor of electricity.

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It is simple. Graphite anode is used because when carbon react with oxygen liberated at anode it forms $\ce{CO2}$ gas which is a exothermic reaction and maintain the temperature of process and not allow molten aluminum to solidify. Because the density of aluminum in solid form is less dense than cryolite and hence it can't be able to extract from the bottom.

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Carbon is a better oxidizing agent than oxygen which produces more heat to retain steady temperature

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